What follows is an essay I wrote about five years ago. I’ll let it speak for itself — but it’s quite a bit longer than my usual posts, so I decided to separate it into two installments. Here is the first part….
(Note: References are to when the essay was written, so that “I currently teach…..” refers to what I was teaching when I wrote the essay.)
What is mathematical creativity?
Forgive me for not answering this question. Better minds have attempted to do so, but no consensus has been reached. I am not confident that a definitive answer will be forthcoming any time soon.
So why ask?
Now this is an interesting question! Perhaps even answerable.
There are those who seek to quantitatively measure creativity in some limited way — but I am not among them. Nor am I convinced that this is a worthwhile endeavor. Of course you will agree with me or not — and I am fairly certain I will not sway you with a few hastily written paragraphs.
We might instead attempt to qualitatively describe mathematical creativity. To what end? Perhaps we might arrange for a team of educators to individually write condensed paragraphs about creativity, but then what is to be done with all the diverse responses? Certainly many such paragraphs have been written already. Consensus is still lacking.
Should I withdraw the question?
Allow me a tentative rewrite. Perhaps, “How might we foster mathematical creativity?”
Much better! But why? We could find an answer potentially useful. Knowing what one teacher did successfully in his classroom could give a colleague an idea which she can adapt for use in her classroom.
Well, this seems to be a promising beginning! A fruitful exchange of ideas, followed by a suitable adaptation, then finally an enthusiastic implementation, and oops! What went wrong?
Learning is situational; teaching is idiosyncratic. From this there is no escape.
Many of us are familiar with the situation where we have two sections of the same class, and what seemed to work wonders in the earlier section is, somehow, not so wonderful in the later section. Perhaps one section was right before lunch, one after. Or a particularly energetic student in one section was sick that day. Maybe a desperate email from a parent just before the later section is lingering heavily on our mind. Rather more likely, however, is just the fact that there are different students in the sections.
Now add to this inescapable fact — that no two classes have the same students — the additional inescapable fact that you are not your colleague. You bring very different backgrounds to your classrooms. Moreover, in creating the lesson, your colleague likely thought through many potential difficulties, then arrived at something he could truly be excited about — and communicated this enthusiasm and confidence to his students in a way which you could not quite match in your classroom.
Nothing went wrong — unless you expected your experience to be the same as your colleague’s. Fortunately, often times it is sufficiently close, but more frequently than we would like, it is not.
This is simply the usual give-and-take we as teachers experience when we are ourselves creative in the classroom. That a new idea is implemented flawlessly is rare; often many revisions are necessary before we are satisfied with the result. An artist may make several sketches before deciding on a particular composition for a painting. A similar patience is required for artistry in teaching.
This suggests that there is no such thing as a successful curriculum. For success is not derived from the structure of a lesson, no matter how cleverly devised. It should be obvious that teachers must be sufficiently well prepared; but sadly this is often not the case. As I have found from interacting with colleagues from around the world, teachers — especially those working with younger children — have meager backgrounds in mathematics. There is an uneasy tension between insisting that teacher candidates have adequate mathematical experience and the real necessity of having them be certified to teach.
For those of us teaching older students, issues of training in both mathematics and pedagogy are significant. I currently teach at a secondary school for students especially talented in mathematics and science. Some of my colleagues (myself included) had previously taught mathematics at university, while others’ careers primarily involved teaching at the secondary level. It should not come as a surprise that such diverse backgrounds result in different views on mathematical creativity — and what is needed to foster it.
As an example, I currently teach a course entitled Advanced Problem Solving. My approach to fostering creativity? Among other things, I have students write an original problem each week on a topic of their choice.
Now given the nature of the students in this course and the course content, students write problems involving logic, geometry, number theory, probability, recurrence relations, generating functions, and geometrical inversion, among others. I give them relatively little guidance, so that they are free to explore and create. I am moderately successful with this approach.
Would I recommend this approach for a new teacher just out of college? With these topics, I would be hesitant except for the most mathematically proficient teacher.
Does that mean new teachers should forego teaching problem posing until they have more experience? Certainly not. I hope to suggest that my style of fostering creativity in the classroom is intimately related to my background and experience — different teachers will take different approaches. Perhaps more importantly, this particular approach plays to my strengths. And — dare I confess? — I get excited about it.
I suspect that every educator knows precisely what I mean. There are courses you teach, and there are courses you are excited to teach. Likely there is no need to wonder in which courses your students are more receptive.
Content is subordinate to engagement. Again, a few paragraphs will not convince you to favor this position if you do not already — but given my own experience as an educator, I stand by it. I am clearly at my best when both my students and myself are thoroughly engaged in the work at hand…those occasional days when students say, “I can’t believe class is over already!” I wish I had more of them.
To be continued….
Creativity in Mathematics 